Discussion
Let \(r = \bar{y}/\bar{x}\) denote the ratio of the sample means of the variable \(y\) and of the auxiliary variable \(x\). The sample is a SRSWOR sample of size \(n\). Let \(f = n/N\) and \(\bar{X}\) be the population mean of the auxiliary variable \(x\). Let \(R = \bar{Y}/\bar{X}\). By considering \(\mathrm{Cov}(r, \bar{x})\), the bias of \(r\), as an estimator of R, is given by :
Answer: D
The bias of \(r\) is \(-\dfrac{1}{\bar{X}}\mathrm{Cov}(r, \bar{x})\).
Since \(r\bar{x} = \bar{y}\) and \(E(\bar{y}) = \bar{Y}\) under SRSWOR,
\[ \begin{aligned} \mathrm{Cov}(r, \bar{x}) &= E(r\bar{x}) - E(r)E(\bar{x}) \\ &= \bar{Y} - E(r)\bar{X} \end{aligned} \]
Dividing by \(\bar{X}\) gives \(R - E(r) = \mathrm{Cov}(r, \bar{x})/\bar{X}\), so the bias \(E(r) - R = -\mathrm{Cov}(r, \bar{x})/\bar{X}\). This is the exact expression; by the Cauchy-Schwarz inequality it also shows \(|\text{bias}| \le \sigma_{r}\sigma_{\bar{x}}/\bar{X}\), small in large samples.
The other options are pieces of the approximate bias formula \(\tfrac{1 - f}{n}R(C_{x}^{2} - \rho C_{x}C_{y})\), with the factor in brackets missing, so they are not the bias.
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